Abstract
We study the Bethe Ansatz formula for the superconformal index, in the case of 4d \( \mathcal{N} \) = 4 super-Yang-Mills with gauge group SU(N). We observe that not all solutions to the Bethe Ansatz Equations (BAEs) contribute to the index, and thus formulate “reduced BAEs” such that all and only their solutions contribute. We then propose, sharpening a conjecture of Arabi Ardehali et al. [1], that there is a one-to-one correspondence between branches of solutions to the reduced BAEs and vacua of the 4d \( \mathcal{N} \) = 1* theory. We test the proposal in the case of SU(2) and SU(3). In the case of SU(3), we confirm that there is a continuous family of solutions, whose contribution to the index is non-vanishing.
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References
A. Arabi Ardehali, J. Hong and J.T. Liu, Asymptotic growth of the 4d \( \mathcal{N} \) = 4 index and partially deconfined phases, JHEP 07 (2020) 073 [arXiv:1912.04169] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
S.M. Hosseini and A. Zaffaroni, Large N matrix models for 3d \( \mathcal{N} \) = 2 theories: twisted index, free energy and black holes, JHEP 08 (2016) 064 [arXiv:1604.03122] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS4, Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].
S.M. Hosseini, A. Nedelin and A. Zaffaroni, The Cardy limit of the topologically twisted index and black strings in AdS5, JHEP 04 (2017) 014 [arXiv:1611.09374] [INSPIRE].
A. Cabo-Bizet, V.I. Giraldo-Rivera and L.A. Pando Zayas, Microstate counting of AdS4 hyperbolic black hole entropy via the topologically twisted index, JHEP 08 (2017) 023 [arXiv:1701.07893] [INSPIRE].
F. Azzurli, N. Bobev, P.M. Crichigno, V.S. Min and A. Zaffaroni, A universal counting of black hole microstates in AdS4, JHEP 02 (2018) 054 [arXiv:1707.04257] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Passias, Holographic microstate counting for AdS4 black holes in massive IIA supergravity, JHEP 10 (2017) 190 [arXiv:1707.06884] [INSPIRE].
F. Benini, H. Khachatryan and P. Milan, Black hole entropy in massive Type IIA, Class. Quant. Grav. 35 (2018) 035004 [arXiv:1707.06886] [INSPIRE].
S.M. Hosseini, I. Yaakov and A. Zaffaroni, Topologically twisted indices in five dimensions and holography, JHEP 11 (2018) 119 [arXiv:1808.06626] [INSPIRE].
P.M. Crichigno, D. Jain and B. Willett, 5d Partition Functions with A Twist, JHEP 11 (2018) 058 [arXiv:1808.06744] [INSPIRE].
M. Suh, Supersymmetric AdS6 black holes from F(4) gauged supergravity, JHEP 01 (2019) 035 [arXiv:1809.03517] [INSPIRE].
S.M. Hosseini, K. Hristov, A. Passias and A. Zaffaroni, 6D attractors and black hole microstates, JHEP 12 (2018) 001 [arXiv:1809.10685] [INSPIRE].
M. Suh, Supersymmetric AdS6 black holes from matter coupled F(4) gauged supergravity, JHEP 02 (2019) 108 [arXiv:1810.00675] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes, JHEP 10 (2019) 062 [arXiv:1810.11442] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
F. Benini and P. Milan, Black Holes in 4D \( \mathcal{N} \) = 4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [INSPIRE].
M. Honda, Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys. Rev. D 100 (2019) 026008 [arXiv:1901.08091] [INSPIRE].
M. Fluder, S.M. Hosseini and C.F. Uhlemann, Black hole microstate counting in Type IIB from 5d SCFTs, JHEP 05 (2019) 134 [arXiv:1902.05074] [INSPIRE].
A. Arabi Ardehali, Cardy-like asymptotics of the 4d \( \mathcal{N} \) = 4 index and AdS5 blackholes, JHEP 06 (2019) 134 [arXiv:1902.06619] [INSPIRE].
J. Kim, S. Kim and J. Song, A 4d \( \mathcal{N} \) = 1 Cardy Formula, JHEP 01 (2021) 025 [arXiv:1904.03455] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The asymptotic growth of states of the 4d \( \mathcal{N} \) = 1 superconformal index, JHEP 08 (2019) 120 [arXiv:1904.05865] [INSPIRE].
A. Amariti, I. Garozzo and G. Lo Monaco, Entropy function from toric geometry, arXiv:1904.10009 [INSPIRE].
D. Gang, N. Kim and L.A. Pando Zayas, Precision Microstate Counting for the Entropy of Wrapped M5-branes, JHEP 03 (2020) 164 [arXiv:1905.01559] [INSPIRE].
G. Kántor, C. Papageorgakis and P. Richmond, AdS7 black-hole entropy and 5D \( \mathcal{N} \) = 2 Yang-Mills, JHEP 01 (2020) 017 [arXiv:1907.02923] [INSPIRE].
A. González Lezcano and L.A. Pando Zayas, Microstate counting via Bethe Ansätze in the 4d \( \mathcal{N} \) = 1 superconformal index, JHEP 03 (2020) 088 [arXiv:1907.12841] [INSPIRE].
A. Lanir, A. Nedelin and O. Sela, Black hole entropy function for toric theories via Bethe Ansatz, JHEP 04 (2020) 091 [arXiv:1908.01737] [INSPIRE].
S. Choi, C. Hwang and S. Kim, Quantum vortices, M2-branes and black holes, arXiv:1908.02470 [INSPIRE].
N. Bobev and P.M. Crichigno, Universal spinning black holes and theories of class ℛ, JHEP 12 (2019) 054 [arXiv:1909.05873] [INSPIRE].
J. Nian and L.A. Pando Zayas, Microscopic entropy of rotating electrically charged AdS4 black holes from field theory localization, JHEP 03 (2020) 081 [arXiv:1909.07943] [INSPIRE].
A. Cabo-Bizet and S. Murthy, Supersymmetric phases of 4d \( \mathcal{N} \) = 4 SYM at large \( \mathcal{N} \), JHEP 09 (2020) 184 [arXiv:1909.09597] [INSPIRE].
F. Benini, D. Gang and L.A. Pando Zayas, Rotating Black Hole Entropy from M5 Branes, JHEP 03 (2020) 057 [arXiv:1909.11612] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The large-N limit of the 4d \( \mathcal{N} \) = 1 superconformal index, JHEP 11 (2020) 150 [arXiv:2005.10654] [INSPIRE].
S. Murthy, The growth of the \( \frac{1}{16} \)-BPS index in 4d \( \mathcal{N} \) = 4 SYM, arXiv:2005.10843 [INSPIRE].
P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong, AdS black holes and finite N indices, arXiv:2005.11240 [INSPIRE].
F. Benini, E. Colombo, S. Soltani, A. Zaffaroni and Z. Zhang, Superconformal indices at large N and the entropy of AdS5 × SE5 black holes, Class. Quant. Grav. 37 (2020) 215021 [arXiv:2005.12308] [INSPIRE].
A. González Lezcano, J. Hong, J.T. Liu and L.A. Pando Zayas, Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions, JHEP 01 (2021) 001 [arXiv:2007.12604] [INSPIRE].
C. Copetti, A. Grassi, Z. Komargodski and L. Tizzano, Delayed Deconfinement and the Hawking-Page Transition, arXiv:2008.04950 [INSPIRE].
K. Goldstein, V. Jejjala, Y. Lei, S. Van Leuven and W. Li, Residues, modularity, and the Cardy limit of the 4d \( \mathcal{N} \) = 4 superconformal index, JHEP 04 (2021) 216 [arXiv:2011.06605] [INSPIRE].
A. Cabo-Bizet, From multi-gravitons to Black holes: The role of complex saddles, arXiv:2012.04815 [INSPIRE].
A. Amariti, M. Fazzi and A. Segati, The superconformal index of \( \mathcal{N} \) = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral, arXiv:2012.15208 [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn — deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
O. Aharony, F. Benini, O. Mamroud and P. Milan, A gravity interpretation for the Bethe Ansatz expansion of the \( \mathcal{N} \)=4 SYM index, [arXiv:2104.13932].
O. Aharony, A gravity interpretation for the Bethe Ansatz expansion of the \( \mathcal{N} \)=4 SYM index, talk given at Fall Seminar Series: Supersymmetric Black Holes, Holography and Microstate Counting, 12 November (2020) http://scgp.stonybrook.edu/video_portal/video.php?id=4628.
C. Closset, H. Kim and B. Willett, \( \mathcal{N} \) = 1 supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [arXiv:1707.05774] [INSPIRE].
F. Benini and P. Milan, A Bethe Ansatz type formula for the superconformal index, Commun. Math. Phys. 376 (2020) 1413 [arXiv:1811.04107] [INSPIRE].
J. Hong and J.T. Liu, The topologically twisted index of \( \mathcal{N} \) = 4 super-Yang-Mills on T2 × S2 and the elliptic genus, JHEP 07 (2018) 018 [arXiv:1804.04592] [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
G. Rizi, Bethe Ansatz formulation for the superconformal index of small N gauge theories, Master’s thesis, University of Trento, Italy (2020). https://www5.unitn.it/Biblioteca/it/Web/TesiDocente/195002.
A.G. Lezcano, J. Hong, J.T. Liu and L.A.P. Zayas, The Bethe-Ansatz approach to the \( \mathcal{N} \) = 4 superconformal index at finite rank, arXiv:2101.12233 [INSPIRE].
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
A. Bourget and J. Troost, The Arithmetic of Supersymmetric Vacua, JHEP 07 (2016) 036 [arXiv:1606.01022] [INSPIRE].
A. Bourget and J. Troost, Counting the Massive Vacua of N = 1* Super Yang-Mills Theory, JHEP 08 (2015) 106 [arXiv:1506.03222] [INSPIRE].
S. Kharchev and A. Zabrodin, Theta vocabulary I, J. Geom. Phys. 94 (2015) 19, [arXiv:1502.04603].
N. Dorey, An Elliptic superpotential for softly broken N = 4 supersymmetric Yang-Mills theory, JHEP 07 (1999) 021 [hep-th/9906011] [INSPIRE].
A. Bourget and J. Troost, On the \( \mathcal{N} \) = 1* gauge theory on a circle and elliptic integrable systems, JHEP 01 (2016) 097 [arXiv:1511.03116] [INSPIRE].
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge University Press (1996) [DOI].
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Benini, F., Rizi, G. Superconformal index of low-rank gauge theories via the Bethe Ansatz. J. High Energ. Phys. 2021, 61 (2021). https://doi.org/10.1007/JHEP05(2021)061
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DOI: https://doi.org/10.1007/JHEP05(2021)061